I think it's perfectly clear that there is a straightforward and obvious interpretation of the question, which results in the answer of 2/3. There are also many strained interpretations, which result in the answer the reader wants to be right.

Again, much like the Monty Haul problem.

So if you meet someone and you are told that they have at least one girl the probability that they have a girl and a boy is 2/3, because the question has one straightforward and obvious interpretation.

And if you meet someone and you are told that they have at least one boy the probability that they have a girl and a boy is 2/3, because the question has one straightforward and obvious interpretation.

And if you meet several people and you are told that they have at least one girl the probability that they have a girl and a boy is always 2/3, because each time the question has one straightforward and obvious interpretation.

And if you meet several people and you are told that they have at least one boy the probability that they have a girl and a boy is always 2/3, because each time the question has one straightforward and obvious interpretation.

And if you meet several people and sometimes you are told that they have at least one boy and sometimes you are told that they have at least one girl the probability that they have a girl and a boy is always 2/3, because each time the question has one straightforward and obvious interpretation.

It's fine to make whatever assumption you need to get the answer you want but that doesn't make it the "straightforward and obvious interpretation". Assume your assumptions!